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Let $n$ be a positive integer $\geqslant 2$ and $\mathbb{N}^n_{\leqslant}$ bethe $n$-th power of positive integers with the product order of the usual orderon $\mathbb{N}$. In the paper we study the semigroup of injective partialmonotone selfmaps of $\mathbb{N}^n_{\leqslant}$ with cofinite domains andimages. We show that the group of units $H(\mathbb{I})$ of the semigroup$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$ is isomorphic to thegroup $\mathscr{S}_n$ of permutations of an $n$-element set, and describe thesubsemigroup of idempotents of$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^n_{\leqslant})$. Also in the case $n=3$we describe the property of elements of the semigroup$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$ as partial bijections ofthe poset $\mathbb{N}^3_{\leqslant}$ and Green's relations on the semigroup$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$. In particular we showthat $\mathscr{D}=\mathscr{J}$ in$\mathscr{P\!O}\!_{\infty}(\mathbb{N}^3_{\leqslant})$.

The monoid of monotone injective partial selfmaps of the poset $(\mathbb{N}^{3},\leqslant)$ with cofinite domains and images